Semi-Smooth Newton Algorithm for Non-Convex Penalized Linear Regression

Both the smoothly clipped absolute deviation (SCAD) and the minimax concave penalty (MCP) penalized linear regression models are capable of dealing with variable selection and parameter estimation simultaneously. Theoretically, these two models enjoy the oracle property even in the high dimensional settings where the number of predictors $p$ may be much larger than the number of observations $n$. However, numerically, it is quite challenging to develop fast and stable algorithms due to their non-convexity and non-smoothness. In this paper we develop a fast algorithm for SCAD and MCP penalized problems. First, we derive that the global minimizers of both models are roots of some non-smooth equations. Then, Semi-smooth Newton (SSN) algorithm is employed to solve the equations. We prove the local superlinear convergence of SSN algorithm. Computational complexity analysis demonstrates that the cost of SSN algorithm per iteration is $O(np)$. Combining with the warmstarting technique SSN algorithm can be very efficient. Simulation studies and real data examples show that SSN algorithm greatly outperforms coordinate descent in computational efficiency while reaching comparable accuracy.